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Shape-morphing finds widespread utility, from the deployment of small stents and large solar sails to actuation and propulsion in soft robotics. Origami structures provide a template for shape-morphing, but rules for designing and folding…

Soft Condensed Matter · Physics 2021-10-25 Xiangxin Dang , Fan Feng , Paul Plucinsky , Richard D. James , Huiling Duan , Jianxiang Wang

The field of rigid origami concerns the folding of stiff, inelastic plates of material along crease lines that act like hinges and form a straight-line planar graph, called the crease pattern of the origami. Crease pattern vertices in the…

Metric Geometry · Mathematics 2025-07-22 Thomas C. Hull

Two-dimensional (2D) origami tessellations such as the Miura-ori are often generalized to build three-dimensional (3D) architected materials with sandwich or cellular structures. However, such 3D blocks are densely packed with continuity of…

Soft Condensed Matter · Physics 2025-07-02 Guowei Wayne Tu , Evgueni T. Filipov

Here we describe an ultra-low-cost origami-based approach for large-scale manufacturing of microscopes, specifically demonstrating brightfield, darkfield, and fluorescence microscopes. Merging principles of optical design with origami…

Optics · Physics 2015-06-19 James Cybulski , James Clements , Manu Prakash

Rigid origami is a branch of origami with great potential in engineering applications to deal with rigid-panel folding. One of the challenges is to compactly fold the polyhedra made from rigid facets with a single degree of freedom. In this…

Applied Physics · Physics 2020-05-15 Yuanqing Gu , Yan Chen

Origami structures are characterized by a network of folds and vertices joining unbendable plates. For applications to mechanical design and self-folding structures, it is essential to understand the interplay between the set of folds in…

Soft Condensed Matter · Physics 2018-03-01 Bryan Gin-ge Chen , Christian D. Santangelo

We develop a theory of random flat-foldable origami. Given a crease pattern, we consider a uniformly random assignment of mountain and valley creases, conditioned on the assignment being flat-foldable at each vertex. A natural method to…

Probability · Mathematics 2025-02-07 Thomas C. Hull , Marcus Michelen , Corrine Yap

Miura-ori is well-known for its capability of flatly folding a sheet of paper through a tessellated crease pattern made of repeating parallelograms. Many potential applications have been based on the Miura-ori and its primary variations.…

Metric Geometry · Mathematics 2020-04-09 Zeyuan He , Simon D. Guest

A strip of square stamps can be folded in many ways such that all of the stamps are stacked in a single pile in the folded state. The stamp folding problem asks for the number of such foldings and has previously been studied extensively. We…

Combinatorics · Mathematics 2025-10-15 Thomas C. Hull , Adham Ibrahim , Jacob Paltrowitz , Natalya Ter-Saakov , Grace Wang

We study the three-dimensional equilibrium shape of a shell formed by a deployed accordion-like origami, made from an elastic sheet decorated by a series of parallel creases crossed by a central longitudinal crease. Surprisingly, while the…

Soft Condensed Matter · Physics 2021-02-09 Théo Jules , Frédéric Lechenault , Mokhtar Adda-Bedia

Thick origami structures are considered here as assemblies of polygonal panels hinged to each other along their edges according to a corresponding origami crease pattern. The determination of the internal actions caused by external loads in…

Soft Condensed Matter · Physics 2024-05-21 Andrea Micheletti , Alessandro Tiero , Giuseppe Tomassetti

This paper considers an extension of origami geometry to the case of "folding" a three dimensional (3D) space along a plane. First, all possible incidence constraints between given points, lines and planes are analyzed by using the geometry…

History and Overview · Mathematics 2018-09-18 Jorge C. Lucero

Non-Euclidean origami is a promising technique for designing multistable deployable structures folded from nonplanar developable surfaces. The impossibility of flat foldability inherent to non-Euclidean origami results in two disconnected…

Soft Condensed Matter · Physics 2023-07-20 Clark Addis , Salvador Rojas , Andres F. Arrieta

We prove that the pleated hyperbolic paraboloid, a familiar origami model known since 1927, in fact cannot be folded with the standard crease pattern in the standard mathematical model of zero-thickness paper. In contrast, we show that the…

Computational Geometry · Computer Science 2009-06-26 Erik D. Demaine , Martin L. Demaine , Vi Hart , Gregory N. Price , Tomohiro Tachi

We develop recursion equations to describe the three-dimensional shape of a sheet upon which a series of concentric curved folds have been inscribed. In the case of no stretching outside the fold, the three-dimensional shape of a single…

Soft Condensed Matter · Physics 2012-12-17 Marcelo A. Dias , Christian D. Santangelo

This article reviews the so-called "axioms" of origami (paper folding), which are elementary single-fold operations to achieve incidences between points and lines in a sheet of paper. The geometry of reflections is applied, and exhaustive…

History and Overview · Mathematics 2017-06-09 Jorge C. Lucero

Traditional origami structures can be continuously deformed back to a flat sheet of paper, while traditional kirigami requires glue or seams in order to maintain its rigidity. In the former, non-trivial geometry can be created through…

Materials Science · Physics 2020-01-29 Xinyu Wang , Simon D. Guest , Randall D. Kamien

Origami structures enabled by folding and unfolding can create complex 3D shapes. However, even a small 3D shape can have large 2D unfoldings. The huge initial dimension of the 2D flattened structure makes fabrication difficult, and defeats…

Computational Geometry · Computer Science 2018-03-12 Zhonghua Xi , Yu-Ki Lee , Young-Joo Lee , Yun-hyeong Kim , Huangxin Wang , Yue Hao , Young-Chang Joo , In-Suk Choi , Jyh-Ming Lien

Inspired by the allure of additive fabrication, we pose the problem of origami design from a new perspective: how can we grow a folded surface in three dimensions from a seed so that it is guaranteed to be isometric to the plane? We solve…

Soft Condensed Matter · Physics 2021-05-19 Levi H. Dudte , Gary P. T. Choi , L. Mahadevan

Origami is the art of paper folding, and it borrows its name from two Japanese words \emph{ori} and \emph{kami}. In Japanese, {ori} means folding, and the paper is called {kami}. While origami is just a hobby to most, there is a lot more to…

History and Overview · Mathematics 2025-03-18 Archana S. Morye